Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-1429051787105x-657535531305401103\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-1429051787105xz^2-657535531305401103z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-22864828593675x-42082296868374264250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2758069/4, 2758065/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 432450 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}$ |
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| Discriminant: | $\Delta$ | = | $518564599414658766523524375000000$ | = | $2^{6} \cdot 3^{22} \cdot 5^{10} \cdot 31^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{1071679233972039583519}{1721868840000} \) | = | $2^{-6} \cdot 3^{-16} \cdot 5^{-4} \cdot 17^{3} \cdot 193^{3} \cdot 3119^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $5.5418149768206182979100752281$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6122994729056535804651996996$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0649290401717362$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.365035433358606$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.0043665416268061205557495040305$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ ( 2 \cdot 3 )\cdot2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $5.2398499521673446668994048366 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $25$ = $5^2$ (exact) |
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BSD formula
$$\begin{aligned} 5.239849952 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{25 \cdot 0.004367 \cdot 1.000000 \cdot 192}{2^2} \\ & \approx 5.239849952\end{aligned}$$
Modular invariants
Modular form 432450.2.a.fy
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5851054080 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $3$ | $4$ | $I_{16}^{*}$ | additive | -1 | 2 | 22 | 16 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $31$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2B | 8.48.0.204 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 248 = 2^{3} \cdot 31 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 8 \\ 172 & 201 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 244 & 243 \end{array}\right),\left(\begin{array}{rr} 24 & 5 \\ 91 & 32 \end{array}\right),\left(\begin{array}{rr} 8 & 63 \\ 247 & 0 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 244 & 245 \end{array}\right),\left(\begin{array}{rr} 241 & 8 \\ 240 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[248])$ is a degree-$14284800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/248\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \) |
| $3$ | additive | $8$ | \( 24025 = 5^{2} \cdot 31^{2} \) |
| $5$ | additive | $18$ | \( 17298 = 2 \cdot 3^{2} \cdot 31^{2} \) |
| $31$ | additive | $272$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 432450.fy
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 28830.y1, its twist by $465$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.